1912]
ELECTRIC CURRENTS IN A THIN ANCHOR-EING
109
The peculiarity of the problem last considered is that the primary current occasions no magnetic force at the surface of the ring. The consequences were set out 40 years ago by Maxwell in a passage* whose significance was very slowly appreciated. " In the case of a current sheet of no resistance, the surface integral of magnetic induction remains constant at every point of the current sheet.
" If, therefore, by the motion of magnets or variations of currents in the neighbourhood, the magnetic field is in any way altered, electric currents will be set up in the current sheet, such that their magnetic effect, combined with that of the magnets or currents in the field, will maintain the normal component of magnetic induction at every point of the sheet unchanged. If at first there is no magnetic action, and no currents in the sheet, then the normal component of magnetic induction will always be zero at every point of the sheet.
"The sheet may therefore be regarded as impervious to magnetic induction, and the lines of magnetic induction will be deflected by the sheet exactly in the same way as the lines of flow of an electric current in an infinite and uniform conducting mass would be deflected by the introduction of a sheet of the same form made of a substance of infinite resistance.
" If the sheet forms a closed or an infinite surface, no magnetic actions which may take place on one side of the sheet will produce any magnetic effect on the other side."
All that Maxwell says of a current sheet is, of course, applicable to the surface of a perfectly conducting solid, such as our anchor-ring may be supposed to be. The currents left in the ring after the abolition of the primary current must be such that the magnetic force due to them is wholly
tangential, to the surface of the ring. Under this condition I M12dd>2 must J J-* r
be independent of <j>1} and we might have investigated the problem upon this basis.
In Maxwell's notation a, /3, 7 denote the components of magnetic force, and the whole energy of the field T is given by
(45)
Moreover «0,the total current, multiplied by 4nr is equal to the "circulation" of magnetic force round the ring. In this form our result admits of immediate application to the hydrodynamical problem of the circulation of
* Electricity and Magnetism, §§ 654, 655. Compare my Mag. 1882, Vol. xm. p. 340 ; Scientific Papers, Vol. n. p. 99.
Acoustical Observations," Phil.thus to allow reflexion in spite of equality of wave- velocities for a given ray.